The plan of this paper is as follows. In section , we show that all graph polynomials for the simplex integral () can be constructed from the corresponding Gram matrix.
The standard parametric representations for Feynman integrals then follow.
Alternatively, by regarding the Schwinger parameters as resistances in an electrical network, we can compute the effective resistances between all vertices of the simplex. This latter set of variables dramatically simplifies the structure of the Schwinger exponential. In section , we use these effective resistances to construct new \(d\rightarrow d+2\)  shift operators for the general \(n\) -point function. The cases \(n=3,4\)  are discussed in detail, and we verify
the action of all operators independently
through computation of their intertwining relations with the conformal Ward identities.
The actions of
the \(d\) -preserving weight-shifting operators of [1]} are also demonstrated from this scalar parametric perspective.
In section , we prove that the new parametric representations
indeed solve the conformal Ward identities. In contrast to the vectorial representation () (for which the Ward identities are analysed in [2]}, [3]}),
for the new scalar parametric representations the Ward identities can be verified directly without use of recursive arguments in the number of points \(n\) .
As we show in section ,
the validity of the conformal Ward identities, as well as the action of the \(d\) -preserving weight-shifting operators, can also be seen from the position-space counterpart of the simplex. Section  concludes with a summary of results and open directions.
